What are the laws of physics? Resisting reification Carlton M. Caves C. M. Caves, C. A. Fuchs, and R. Schack, “Subjective probability and quantum certainty,”

What are the laws of physics?Resisting reification

Carlton M. CavesC. M. Caves, C. A. Fuchs, and R. Schack, “Subjective probability and quantum certainty,”

Studies in History and Philosophy of Modern Physics 38, 255-274 (2007).

C. G. Timpson, “Quantum Bayesianism: A Study,” Studies in History and Philosophy of Modern Physics 39, 579-609 (2008).

Department of Physics and AstronomyUniversity of New Mexico

[email protected]://info.phys.unm.edu/~caves

The laws are out there. Probabilities aren’t.

mailto:[email protected]

http://info.phys.unm.edu/~caves

http://info.phys.unm.edu/~caves

Some mathematical objects in a scientific theory are our tools; others correspond to reality. Which

is which?

Laws of physics?

Oljeto Wash Southern Utah

Subjective Bayesian probabilities

QM: Derivation of quantum probability rule from infinite frequencies?

Objective probabilities

● Logical probabilities (objective Bayesian): physical symmetry implies probability

● Probabilities as frequencies: probability as verifiable fact

● Objective chance (propensity): probability as specified fact

■ Symmetries are applied to judgments, not to facts.

■ Probabilities are used routinely for individual systems.■ Frequencies are observed facts, not probabilities. ■ Bigger sample space: exchangeability.

■ Some probabilities are ignorance probabilities, but others are specified by the facts of a “chance situation.”■ Specification of “chance situation”: same, but different.

objective chance

QM: Probabilities from physical law. Salvation of objective chance?

C. M. Caves, R. Schack, ``Properties of the frequency operator do not imply the quantum probability postulate,'' Annals of Physics 315, 123-146 (2005) [Corrigendum: 321, 504--505 (2006)].


Facts

Outcomes of eventsTruth values of propositions

Objective

Probabilities

Agent’s degree of beliefin outcome of an event or

truth of a proposition

Subjective

Facts never imply (nontrivial) probabilities.

Two agents in possession of the same facts can assign different probabilities.

Category distinction


Probabilities

Agent’s degree of belief in outcome of an event or truth of a proposition.

Consequence of ignorance

Agent’s betting odds

Subjective

Agent A regards $q as fair price for the

ticket.

A assigns p(E)=q.

Dutch-book consistency

The standard rules for manipulating probabilities are objective consequences of

requiring consistent betting behavior.

A’s probability assignments, i.e., ticket prices, are inconsistent if they can lead to a sure

loss.

The usual argument: If A does not obey the probability rules, she will lose in the long run.

Dutch-book argument: If A does not obey the probability rules, she will lose in one shot.

Dutch-book argument: Rules (i) and (ii)

A is willing to sell ticket for a negative amount. Sure loss.

A is willing to sell ticket, which is definitely worth $1 to her, for less than $1.Sure loss.

Dutch-book argument: Rule (iii)

A would buy the purple ticket for $q and sell the green tickets for $r + $s. If q > r + s, sure loss.

Dutch-book argument: Rule (iv)


The standard rules of probability theory are objective

consequences of requiring consistent betting behavior.


Facts in the form of observed data d are used to update probabilities via Bayes’s rule:

posterior

prior

conditional (model, likelihood)

The posterior always depends on the prior, except when d logically implies h0:

Facts never determine (nontrivial) probabilities.

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Are quantum probabilities subjective?

Classical (realistic, deterministic) world

Quantum world

State spaceSimplex of probabilities for

microstates Convex set of density operators

StateExtreme point

MicrostateEnsemble

Extreme point

Pure state

State vector

Ensemble

Mixed state

Density operator

Objective Subjective Objective Subjective

Scorecard:1. Predictions for fine-grained measurements2.Verification (state determination)3.State change on measurement4.Uniqueness of ensembles5.Nonlocal state change (steering)6.Specification (state preparation)

Certainty:


Quantum world



StateExtreme point

MicrostateEnsemble

Extreme point

Pure state

State vector

Ensemble

Mixed state

Density operator

Fine-grained measurement

Certainty ProbabilitiesCertainty or

ProbabilitiesProbabilities


Whom do you ask for the system state? The system or an agent?


Quantum world



StateExtreme point

MicrostateEnsemble

Extreme point

Pure state

State vector

Ensemble

Mixed state

Density operator

Verification:

state determinationYes No No No


Quantum world



StateExtreme point

MicrostateEnsemble

Extreme point

Pure state

State vector

Ensemble

Mixed state

Density operator

Can you reliably distinguish two nonidentical states?

iff orthogonal

Alwaysiff orthogonal iff orthogonal iff orthogonal


Quantum world



StateExtreme point

MicrostateEnsemble

Extreme point

Pure state

State vector

Ensemble

Mixed state

Density operator

Can you unambiguously distinguish two nonidentical states?

Always

Sometimes

(iff supports not identical)

Always

(supports are not identical)

Sometimes

(iff supports not identical)


Quantum world



StateExtreme point

MicrostateEnsemble

Extreme point

Pure state

State vector

Ensemble

Mixed state

Density operator

Verification:



Whom do you ask for the system state? The system or an agent?


Quantum world



StateExtreme point

MicrostateEnsemble

Extreme point

Pure state

State vector

Ensemble

Mixed state

Density operator

State change on measurement

No Yes Yes Yes

State-vector reduction or wave-function collapse

Real physical disturbance?

Objective Subjective Subjective Subjective


Quantum world



StateExtreme point

MicrostateEnsemble

Extreme point

Pure state

State vector

Ensemble

Mixed state

Density operator

Uniqueness of ensembles

Yes No No No



Quantum world



StateExtreme point

MicrostateEnsemble

Extreme point

Pure state

State vector

Ensemble

Mixed state

Density operator

Nonlocal state change (steering)

No Yes Yes Yes


Real nonlocal physical disturbance?


Quantum world



StateExtreme point

MicrostateEnsemble

Extreme point

Pure state

State vector

Ensemble

Mixed state

Density operator




Verification:



No Yes Yes Yes


Yes No No No


No Yes Yes Yes

Specification:

state preparationYes No ? ?


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Northern New Mexico

Copenhagen vs. Bayes


Quantum world



StateExtreme point

MicrostateEnsemble

Extreme point

Pure state

State vector

Ensemble

Mixed state

Density operator

Specification:

state preparationYes No Copenhagen: Yes Copenhagen: Yes

Copenhagen interpretation: Classical facts specifying

the properties of the preparation device

determine a pure state.

Objective Subjective Objective Objective

Copenhagen (objective preparations view) becomes

the home of objective chance, with nonlocal physical

disturbances.

Copenhagen


Quantum world



StateExtreme point

MicrostateEnsemble

Extreme point

Pure state

State vector

Ensemble

Mixed state

Density operator




Verification:



No Yes Yes Yes


Yes No No No


No Yes Yes Yes

Specification:

state preparationYes No Yes Yes

Objective Subjective Objective Objective

Classical and quantum updating Facts in the form of observed

data d are used to update probabilities via Bayes’s rule:

posterior

prior

conditional (model, likelihood)

The posterior always depends on the prior, except when d

logically implies h0:The posterior state always depends on

prior beliefs, even for quantum state preparation, because there is a

judgment involved in choosing the quantum operation.

Facts in the form of observed data d are used to update

quantum states:

posterior

prior

quantum operation (model)

Quantum state preparation:

Facts never determine probabilities or quantum

states.

Where does Copenhagen go wrong?

The Copenhagen interpretation forgets that the preparation device is quantum

mechanical. A detailed description of the operation of a preparation device (provably)

involves prior judgments in the form of quantum state assignments.

It is possible to show that neither deterministic nor stochastic preparation devices can

prepare the same system state independent of system and device initial states.

Subjective Bayesian


Quantum world



StateExtreme point

MicrostateEnsemble

Extreme point

Pure state

State vector

Ensemble

Mixed state

Density operator




Verification:



No Yes Yes Yes


Yes No No No


No Yes Yes Yes

Specification:

state preparationYes No No No


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Western Australia

Bayesian quantum probabilities

Quantum states vs. probabilities

Are quantum states the same as probabilities? No, though both are subjective,

there are differences, but these differences should be stated in Bayesian terms.

A quantum state is a catalogue of probabilities, but the rules for manipulating

quantum states are different than for manipulating probabilities.

The rules for manipulating quantum states are objective consequences of restrictions on how

agents interface with the real world.

Catalogue of probabilities: Fuchs’s gold standard

Symmetric Informationally Complete (SIC)-POVM

Cable BeachWestern Australia

Quantum coin tossing

Is a quantum coin toss more random than a classical one?

Why trust a quantum random generator over a classical one?

quantum coin toss


Quantum world



StateExtreme point

MicrostateEnsemble

Extreme point

Pure state

State vector

Ensemble

Mixed state

Density operator




C. M. Caves, R. Schack, “Quantum randomness,” in preparation.

Measure spin along z axis:

Measure spin along x axis:

quantum coin toss

Measure spin along z axis:

Measure spin along x axis:

Standard answer: The quantum coin toss is objective, with probabilities guaranteed by physical law.

Subjective Bayesian answer? No inside information.

Is a quantum coin toss more random than a classical one?

Why trust a quantum random generator over a classical one?

Pure states and inside information

Party B has inside information about event E, relative to party A, if A is willing to agree to a bet on E that B believes to be a sure win. B has one-way inside information if B has inside information relative to A, but A does not have any inside information relative to A.

The unique situation in which no other party can have one-way inside information relative to a party Z is when Z assigns a pure state. Z is said to have a maximal belief structure.

Subjective Bayesian answerWe trust quantum over classical coin tossing because an agent who believes the coin is fair cannot rule out an insider attack, whereas the beliefs that lead to a pure-state assignment are inconsistent with any other party’s being able to launch an insider attack.

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A stab at ontology

A stab at ontology

Quantum systems are defined by attributes, such as position, momentum, angular momentum, and energy or Hamiltonian. These attributes—and thus the numerical particulars of their eigenvalues and eigenfunctions—are objective properties of the system.

The value assumed by an attribute is not an objective property, and the quantum state that we use to describe the system is purely subjective.

A stab at ontology

1. The attributes orient and give structure to a system’s Hilbert space. Without them we are clueless as to how to manipulate and interact with a system.

2. The attributes are unchanging properties of a system, which can be determined from observable facts. The attributes determine the structure of the world.

3. The system Hamiltonian is one of the attributes, playing the special role of orienting a system’s Hilbert space now with the same space later.

4. Convex combinations of Hamiltonian evolutions are essentially unique (up to degeneracies).

Why should you (I) care?If you do care, how can this be made convincing?

Status of quantum operations?Effective attributes and effective Hamiltonians? “Effective reality”?

Kookaburras in New Mexico

What are the laws of physics? Resisting reification Carlton M. Caves C. M. Caves, C. A. Fuchs, and R. Schack, “Subjective probability and quantum certainty,”

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